Let $G$ be a $k$-partite $k$-uniform hypergraph with at least $dn^k$ many edges. I want a lower bound on the number of $K_{2, 2, ...,2}$ in $G$, preferably something like $\gamma n^{2k}$ for some constant $\gamma$ independent of $n$.
My ideas: I can get down to a minimum degree subhypergraph from the edge-density by cleaning up vertices with low degrees. Starting from the edge-density I can also get down to a hypergraph where all vertices have a partite minimum co-degree either $0$ or $\beta n^{k-1}$. But I have no idea of how to do both of that together. I have also been thinking of some kind of induction approach but have not been able to prove it with that.